Problem: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{r^2 - 36}{r - 6}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = r$ $ b = \sqrt{36} = -6$ So we can rewrite the expression as: $z = \dfrac{({r} {-6})({r} + {6})} {r - 6} $ We can divide the numerator and denominator by $(r - 6)$ on condition that $r \neq 6$ Therefore $z = r + 6; r \neq 6$